There are a few.
Selenium as mentioned above is one of the most popular.
However, there are also other UI tools that work just as well with web interfaces. It sounds like you're trying to avoid coding, but if I'm wrong:
Coding ones
WaTiR - Web Automated Testing in Ruby - that's a very popular one that handles any interface on a web page, aside from interacting with java applets directly. WaTiJ exists as well if you prefer Java, and WaTiN if you like .NET. There are plenty more but I thought I'd mention that in particular, for the rest, check:
http://www.softwareqatest.com/qatweb1.html
Performance
While I'm here, I thought I'd mention the previous link again if you're looking to thrash your app heavily - there are a bunch of load tools there too.
Non-coding
This is probably what you're after. Selenium as mentioned before, DOES have coding, but also has a record-and-playback feature. So (in theory) you record yourself doing the test, and it can reproduce it reliably hundreds of times afterwards :) There are other tools that work in regular and web apps - AutoIt, Rational Robot, Mercury Tools and more, but Selenium is the most popular.
Required reading
If you do go with Selenium, I recommend the following reading
http://www.slideshare.net/joaopmaia/web-app-testing-with-selenium - a slideshow on how to use Selenium for web testing
http://www.pushtotest.com/docs/testmaker-help/testmaker-tutorials/building-tests-of-rich-internet-applications-using-ajax-and-selenium/task-1-record-and-playback-a-web-application-functional-test - a sample of how to use Selenium IDE's record and playback features.
Good luck!